Mass transport and uniform rectifiability

In this paper we characterize the so called uniformly rectifiable sets of David and Semmes in terms of the Wasserstein distance $W_2$ from optimal mass transport. To obtain this result, we first prove a localization theorem for the distance $W_2$ which asserts that if $μ$ and $ν$ are probability measures in $R^n$, $ϕ$ is a radial bump function smooth enough so that $\intϕdμ\gtrsim1$, and $μ$ has a density bounded from above and from below on the support of ϕ, then $W_2(ϕμ,aϕν)\leq c W_2(μ,ν),$ where $a=\intϕdμ/ \intϕ\,dν$.

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